Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

1. Problem Statement

The paper investigates the Cauchy problem for a heat equation governed by a mixed local–nonlocal diffusion operator with spatially irregular (distributional) coefficients. The equation is given by:

$$\begin{cases} \partial_t u(t, x) + L u(t, x) = 0, & (t, x) \in (0, T) \times \mathbb{R}^d, \\ u(0, x) = u_0(x), & x \in \mathbb{R}^d, \end{cases}$$

where the operator $L$ is defined as:
$$L u = -\nabla \cdot (a(x) \nabla u) + (-\Delta)^{s/2} \left( b(x) (-\Delta)^{s/2} u \right) + c(x) u,$$
with $0 < s < 1$.

Key Challenges:

• Mixed Nature: The operator combines a local diffusion term ($-\nabla \cdot (a \nabla u)$) and a nonlocal fractional diffusion term involving the Riesz fractional Laplacian $(-\Delta)^{s/2}$.
• Irregular Coefficients: The coefficients $a(x), b(x), c(x)$ and the initial data $u_0(x)$ are assumed to be distributions ($u_0, a, b, c \in \mathcal{D}'(\mathbb{R}^d)$), not necessarily functions.
• The Obstruction: In the classical theory of partial differential equations (PDEs), the product of distributions is generally undefined (Schwartz's impossibility result). Therefore, the equation cannot be interpreted in the classical or standard weak sense when coefficients are singular.

2. Methodology

To overcome the obstruction of multiplying distributions, the authors employ the theory of very weak solutions, originally introduced by Garetto and Ruzhansky. The methodology proceeds in three main stages:

A. Regularization (Friedrichs Mollification)

The singular coefficients and initial data are regularized using a Friedrichs mollifier $\psi_{\omega(\varepsilon)}$ (a standard smooth approximation of the identity with scale $\omega(\varepsilon) \to 0$ as $\varepsilon \to 0$).

• Define regularized nets: $a_\varepsilon = a * \psi_{\omega(\varepsilon)}$, $b_\varepsilon = b * \psi_{\omega(\varepsilon)}$, etc.
• This transforms the singular problem into a family of regularized problems with smooth coefficients:
  $$\partial_t u_\varepsilon + L_\varepsilon u_\varepsilon = 0, \quad u_\varepsilon(0) = u_{0,\varepsilon}.$$

B. Classical Well-Posedness for Regularized Problems

For each fixed $\varepsilon > 0$, the coefficients are smooth ($C^\infty$) and bounded. The authors establish classical well-posedness for these regularized problems using:

• Energy Methods: Defining a bilinear form $B_\varepsilon(u, v)$ associated with $L_\varepsilon$.
• Galerkin Approximation: Constructing approximate solutions in finite-dimensional subspaces.
• A Priori Estimates: Deriving uniform energy estimates in the space $L^\infty(0, T; H^1(\mathbb{R}^d)) \cap C([0, T]; L^2(\mathbb{R}^d))$.

C. Very Weak Solution Framework

The solution to the original singular problem is defined as the limit of the net of regularized solutions $(u_\varepsilon)_\varepsilon$.

• Moderateness: The nets of coefficients and solutions must satisfy "moderateness" conditions (growth bounds of the form $C \omega(\varepsilon)^{-N}$ as $\varepsilon \to 0$) to ensure they do not blow up too fast.
• Negligibility: Uniqueness is defined up to "negligible" nets (nets that decay faster than any power of $\omega(\varepsilon)$).
• Consistency: The framework must recover the classical weak solution when the coefficients are actually regular.

3. Key Contributions and Results

1. Classical Well-Posedness (Regular Case)

The authors first prove existence and uniqueness for the case where coefficients $a, b, c \in L^\infty(\mathbb{R}^d)$ are bounded and measurable, satisfying uniform positivity ($a \ge a_0 > 0$, $b \ge b_0 > 0$, $c \ge 0$).

• Result: There exists a unique weak solution $u$ in the energy space.
• Estimate: An a priori estimate is derived:
  $$\|u(t)\|_{L^2}^2 + \|\nabla u(t)\|_{L^2}^2 + \|(-\Delta)^{s/2} u(t)\|_{L^2}^2 \le C \|u_0\|_{H^1}^2.$$
  This ensures continuous dependence on initial data.

2. Existence of Very Weak Solutions

For distributional coefficients $a, b, c, u_0 \in \mathcal{E}'(\mathbb{R}^d)$ (compactly supported distributions):

• Theorem: Under the assumption that the distributions are strictly positive (in the sense of distributions), there exists a very weak solution $(u_\varepsilon)_\varepsilon$.
• Mechanism: The regularized solutions $u_\varepsilon$ form a net that is $L^\infty(0, T; H^1)$-moderate. This means the solution net grows at a controlled rate relative to the regularization parameter $\varepsilon$.

3. Uniqueness of Very Weak Solutions

• Theorem: The very weak solution is unique up to negligibility.
• Detail: If two different regularizations of the same distributional data are used, the difference between the resulting solution nets $(u_\varepsilon - \tilde{u}_\varepsilon)$ is a negligible net in the space $L^\infty(0, T; L^2) \cap L^2(0, T; H^1)$. This implies the solution is independent of the specific choice of mollifier or regularization scale, provided the nets are moderate.

4. Consistency with Classical Theory

• Theorem: If the coefficients and initial data are actually regular (i.e., $a, b, c \in L^\infty$ and $u_0 \in H^1$), the very weak solution converges to the unique classical weak solution.
• Convergence: As $\varepsilon \to 0$, $u_\varepsilon \to u$ in $C([0, T]; L^2(\mathbb{R}^d)) \cap L^2(0, T; H^1(\mathbb{R}^d))$. This validates the very weak solution concept as a genuine extension of the classical theory.

4. Significance and Impact

• Unified Framework for Singular Models: This work provides the first rigorous analytical framework for mixed local–nonlocal parabolic equations with spatially singular coefficients. Previous studies typically assumed smooth or at least bounded coefficients.
• Handling Non-Commutative Products: By utilizing the very weak solution approach, the paper bypasses the fundamental obstruction of multiplying distributions, allowing for the modeling of physical phenomena where material properties (diffusivity, reaction rates) are highly heterogeneous or fractal-like.
• Applications: The results are directly applicable to:
  • Ecology: Modeling animal movement with mixed local diffusion and long-range Lévy jumps in environments with irregular resource distributions.
  • Physics: Anomalous transport in porous media or turbulent flows where local and nonlocal effects coexist with singular heterogeneities.
  • Mathematical Biology: Optimal foraging models where the environment is not smooth.
• Foundation for Future Work: The established energy estimates and consistency results lay the groundwork for future numerical approximations and the study of nonlinear versions of these mixed operators with singular data.

In summary, the paper successfully extends the theory of heat equations to a highly singular setting by combining mixed local-nonlocal operators with the "very weak solution" methodology, ensuring mathematical rigor while preserving physical relevance.